We study pattern formation and report on new instabilities in reaction-advection-diffusion systems of two different kinds.
First, we develop a theory describing the transition to a spatially homogeneous regime in a mixing flow with a chaotic in time reaction . The transverse Lyapunov exponent governing the stability of the homogeneous state can be represented as a combination of Lyapunov exponents for spatial mixing and temporal chaos. This representation, being exact for time-independent flows and equal Peclet numbers of different components, is demonstrated to work accurately for time-dependent flows and different Peclet numbers. The properties of structures that appear beyond the stability threshold are discussed.
Second, we consider a reaction-diffusion system of an activator-inhibitor type and impose the periodic in space mixing flow. We fix the governing parameters in the way that ensures the stability of the homogeneous steady state in reaction-diffusion system, so that without advection no Turing patterns can occur. Next, we increase the advection rate and study the influence of the flow on the stability of this state. One could intuitively expect that because the flow is mixing, it should stabilize the homogeneous state. However, the instability appears as the rate of advection increases beyond a certain threshold, which results in pattern formation. We apply the Bloch theory to find out the length-scale of the patterns, which generally do not coincide with the length-scale of the imposed flow. The mechanism of the instability can be understood from a reduced model; the results are explained by means of Lyapunov exponents. We consider two situations: (i) the general case when both chemical species are advected and (ii) a partial case, when only one species is advected, which is relevant to biological applications.
Since the flow and reaction terms are generic for the effects investigated, we believe that the results hold for a variety of flows and chemical reactions.
 A.V. Straube, M. Abel, A. Pikovsky, Temporal chaos versus spatial mixing in reaction-advection-diffusion systems, Phys. Rev. Lett. 93, 174501 (2004).